The axiom of choice is a common set-theoretic axiom with many equivalents and consequences. This tag is for questions on where we use it in certain proofs, and how things would work without the assumption of this axiom.

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Zorn's lemma converse? (Context: Maximal proper subgroups)

So, in my qual prep class a pretty simple question popped up: "Prove that for any nontrivial finite group there exists a maximal proper subgroup." So of course, my natural inclination was to ...
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$|\alpha\times\alpha|=|\alpha|$ for infinite ordinal $\alpha$, definably

I am trying to prove Proposition 1.7 of http://math.bu.edu/people/aki/7.pdf: Proposition 1.7 (Halbeisen–Shelah). If $\aleph_0\le|X|$, then $|{\cal P}(X)|\not\le|{\rm Seq}(X)|$. In words, ...
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What's wrong with this proof of the inconsistency of the axiom of choice?

Let $\mathscr{T}$ be the (countable) collection of all theorems provable in ZFC. Define an equivalence relation on $\mathscr{T}$ by $\phi\sim\psi$ iff $(\phi \iff \psi)$. In other words, two theorems ...
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Choice function for collection of arbitrary finite sets. AC required?

I understand how we can show the existence of a choice function for any (finite or infinite) collection of (finite or infinite) subsets of, say, $\mathbb{N}$ or $\mathbb{Z}$ without using the axiom of ...
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Looking for explanation of Banach-Tarski Proof, preferably by visual methods “Video, Pictures, Diagrams…”

could someone please explain the four steps of Banach-Tarski? 1- Find a paradoxical decomposition of the free group in two generators. 2- Find a group of rotations in 3-d space isomorphic to the free ...
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Is AC necessary to show that in metric spaces $x\in\operatorname{closure}(A)$ implies $\exists\{a_n\}_{n=1}^\infty\subseteq A$ s.t. $\lim a_n=x$?

Let $(X,d)$ be a metric space. Let $x\in\operatorname{closure}(A)$, where $A\subseteq X$. Then for each $n\in\mathbb{N},\exists x_n\in B_{\frac{1}{n}}(x)\cap A$, where $B_\varepsilon(x)$ is the open ...
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If $|A|<|B|$ does $B$ surject onto $\aleph(A)$?

After reading Proving existence of a surjection $2^{\aleph_0} \to \aleph_1$ without AC I became curious if there is a generalization to arbitrary cardinals. That is, if $\frak m<n$, does it follow ...
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How to show countability of $\omega^\omega$ or $\epsilon_0$ in ZF?

I know that with choice, the countable union of countable sets is countable, making $\omega^\omega$ and $\epsilon_0$ both countable. Can we show this without choice? E.g. in the case that $\omega_1$ ...
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How to prove that $\sf CH$ implies $2^{\aleph_0}=\aleph_1$

Of course, most of you will, upon reading the title, exclaim "But isn't that the definition of the continuum hypothesis?" So I need to be a little more careful about the exact definitions. Let ...
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Does “$(\exists f:A\twoheadrightarrow B)\implies(\exists f:B\hookrightarrow A)$” implies the axiom of choice? [duplicate]

Let $P$ denotes the property that if there exists a surjection from set $A$ to set $B$, then there exists an injection from $B$ to $A$. It's apparent that $P$ can be proved in ZFC. My question is that ...
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Can we prove, without axiom of choice, that the set of all zero divisors (including $0$) of a commutative ring with unity contains a prime ideal?

Let $R$ be a commutative ring with unity , I know that assuming axiom of choice , if $A$ is the set of all zero divisors (including $0$ ) then it is a union of prime ideals so it contains a prime ...
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The existence of the sequence corresponding to some asymptotic sequence

The following proof of the axiom of choice by induction is obviously false: Let $(\Lambda)_{i=1, 2, \ldots}$ be an infinite sequence of nonempty sets. When $i=1$, self-evident. We will assume this ...
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Does a direct sum decomposition of an infinite-dimensional vector space require Zorn's lemma?

Let $V$ be an infinite-dimensional vector space and $V'\subset V$ a subspace. Does it require Zorn's lemma to write $V=V'\oplus V''$ for some other subspace $V''\subset V$?
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Axiom of choice : continuous function and uniformly continuous

How I proof that every continuous function f in [0,1] is uniformly continuous, without axiom o choice? I took this from the book Axiom of Choice from Horst Herrilich He had a observation that ...
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Does the proof of Bolzano-Weierstrass theorem require axiom of choice?

When selecting the terms of subsequence from each bisections, I thought axiom of choice might be required. But I'm not so sure whether or not, so please tell me. [edited] I'm sorry for the lack of ...
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Can we prove AC from the statement “There is no $\aleph$ cardinal strictly between $\operatorname{CARD}(X)$ and $\operatorname{CARD}(2^X)$”?

If $X$ is a set, let $\operatorname{CARD}(X)$ denote the Cardinal number of $X$. Let GCH(1) be the statement "If $K$ is an infinite initial ordinal number, then there exists no initial ordinal number ...
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Is it consistent without the axiom of choice that every permutation of some infinite set have fixed points?

A "permutation" of a non-empty set means an injective mapping of the set onto itself. Let $S(1)$ be the statement "There exists a permutation of every set containing at least two elements, which has ...
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A question about infinite sets and Cantor's Power Set theorem

Let $\operatorname{Card}(X)$ denote the cardinal number of the set $X$. The standard proof of Cantor's Power Set theorem stating that "$\operatorname{Card}(X) < \operatorname{Card}(2^X)$" is ...
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Choosing a Cauchy sequence for a real

It is easy to form in ZF, for each real $a$, a "canonical" Cauchy sequence that converges to $a$. For example, one can take the sequence of finite initial segments of the decimal expansion of $a$, ...
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Axiom of choice for singletons

Let $\mathscr C \subseteq \mathscr P (\mathbb R )$ be a family of singletons, i.e.: each element of $\mathscr C$ contains exactly one real number. Let $f: \mathscr C \to \mathbb R $ be the function ...
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Zorn's Lemma's chain condition

Zorn's Lemma requires that every chain in a partially ordered set $X$ has an upper bound. In this article Gowers uses Zorn's Lemma to find a maximal linearly independent (over $\mathbb{Q}$) subset of ...
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Regularity of $\omega_1$ and axiom of choice

Why is the regularity of the ordinal $\omega_1$ a consequence of the axiom of choice?
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Basis of $\mathbb{F}[[x]]$ over $\mathbb{F}$ without AC

Does the ring of formal power series $\mathbb{F}[[x]]$ as a vector space over $\mathbb{F}$ admit a basis without assuming the Axiom of choice, at least in some special cases of $\mathbb{F}$? I'm ...
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Difference between Zorn's Lemma and the ascending chain condition

Let $S$ be a non-empty partially ordered set with respect to a relation $\leq$. Then: Zorn's Lemma: If $S$ has the property that any totally ordered subset $U\subset S$ has an upper bound, then ...
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Choice of a skeleton

Suppose we are in presence of a strong enough axiom of choice (e.g., choice for conglomerates). I know that any category has a skeleton, but I would like to know if I can choose a skeleton which ...
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In P. Cohen's models (or others) may we have $\neg\mathsf{AC}+\mathsf{CH}$? May we have $\neg \mathsf{AC} + \neg \mathsf{CH}$?

I know we have the consistency of $\mathsf{ZF} + \mathsf{AC} + \mathsf{GCH}$, $\mathsf{ZF} + \neg \mathsf{AC}$, and $\mathsf{ZF} + \mathsf{AC} + \neg \mathsf{GCH}$. What about $\mathsf{ZF} + \neg ...
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Is the axiom of choice really all that important?

According to this book: The Axiom of Choice is the most controversial axiom in the entire history of mathematics. Yet it remains a crucial assumption not only in set theory but equally in modern ...
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Equivalence of countable choice for subsets of the reals and “second countable $\implies$ Lindelöf”

Looking for a proof that in a second countable space every open cover has a countable subcover -- i.e. every s.c. space is a Lindelöf space -- I bumped into this question. That answered my question. I ...
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Prove there exists a strictly increasing function from the natural numbers to a partially strictly ordered set

Let $P$ be a non-empty partially strictly ordered set and assume no element of $P$ is maximal (i.e. for every $x \in P$ there exists $y\in P$ with $x < y$). Show there exists a function ...
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Every II-finite set is III-finite

I need some help proving that if a set $X$ is II-finite then it is III-finite, i.e. if every non-empty family of subsets of $X$ which is linearly ordered by inclusion has a maximal element under ...
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Lebesgue measure without choice

From this question and this question (and their answers) I gather that it is consistent with ZF without The Axiom of Choice to assume that there exist countable sets $A_n$, $n\in \mathbb N$, such that ...
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Accumulation point is a limit of some sequences in first-countable space with Axiom of Choice.

I have a statement which says that Let $X$ be a first-countable space and $E$ be a subset of $X$. If $p$ is an accumulation point of $E$, then there exists a sequence in $E$ that converges to ...
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Finite complement topology over $\Bbb{R}$ is not second-countable under ZF?

Under $\mathsf{ZF+AC_\omega}$, the space $\Bbb{R}$ with finite complement topology is not second-countable, since a countable union of countable subsets of $\Bbb{R}$. However, such argument doesn't ...
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Breaking the AC barrier using Kelley-Morse set theory

In her blog post Variants of Kelley-Morse set theory, Prof. Gitman proves that every model of the the common version of Kelley-Morse set theory ($\mathsf{KM}$) is a model of the Wikipedia version of ...
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Every set with more than point admits a permutation with no fixed point and the Axiom of Choice [duplicate]

Assuming axiom of choice , for any set $S$ with more than one point , there exist a bijection $f:S \to S$ such that $f(s) \ne s , \forall s \in S$ . Is the converse true , i.e. Does the statement " ...
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Consequences of the negation of the Axiom of Dependent Choice

It seems to me that a proper reason to include The Axiom of Choice as a foundational axiom of set theory should be based on the observation that the negation of The Axiom of Choice has absurd ...
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$\forall \alpha \exists \beta: \beta > \alpha$ where $\alpha$ and $\beta$ cardinals

I have to prove ZF $\vdash$ $\forall \alpha \exists \beta:\beta > \alpha$, where $\alpha, \beta-$ cardinal numbers. I can prove it only in ZFC. Let's fix some cardinal number $\alpha$. By ...
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In every ring with unity satisfying ACC, every ideal is finitely generated; can we prove it without assuming Axiom of choice?

Assuming Zorn's lemma, we can prove that in every ring with unity satisfying Ascending-Chain-Condition, every ideal is finitely generated. Is this statement equivalent to Zorn's lemma? Can we prove it ...
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In any commutative ring with unity, every prime ideal is finitely generated implies every ideal is finitely generated; can it be prove without A.C.?

Assuming Zorn's lemma, "In any commutative ring with unity, if every prime ideal is finitely generated, then every ideal is finitely generated". Is the converse true, i.e. if in any commutative ring ...
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Assuming the axiom of choice ,how to prove that every uncountable abelian group must have an uncountable proper subgroup?

Assuming the axiom of choice , how to prove that every uncountable abelian group must have an uncountable proper subgroup ? Related to Does there exist any uncountable group , every proper subgroup ...
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Can you verify this proof of the Schroeder-Bernstein theorem?

I'm a freshman in college and my professor challenged us to find a proof of this theorem. Please don't give me the answer but please verify if this proof works or, if not, if it is the start of a ...
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Proving that “Every non-trivial ring (i.e. with more than one element ) with unity has a maximal ideal” implies axiom of choice is true

I know that assuming axiom of choice or equivalently Zorn's lemma , it can be proved that every non-trivial ring with unity has a maximal ideal (two sided ) . The wiki article on axiom of choice says ...
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Formal proof that $N$ is the smallest infinite set [duplicate]

I wish to write a formal proof of the following statement: For any infinite set $X$, there exists an injection $f:\mathbb{N}\to X$. I'd like the proof to explicitly use the full axiom of choice (for ...
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Acyclicity of Flasque sheaves without A.C.

I say that a sheaf on a space X, is flasque if the restriction maps are surjective, that is any local section extend to a global section. Now it is a fact that if $F_1$ is flasque and if $0 \to F_1 ...
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Cardinality of sets of reals without choice

Assuming just ZF (no axiom of choice): Does $\aleph_n\leq|\mathbb{R}|$ for all $n<\omega$ imply $\aleph_\omega\leq|\mathbb{R}|$? (with $\kappa\leq|\mathbb{R}|$ meaning that there is a set of reals ...
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Conceptual problem in Noetherian rings and Zorn Lemma

I've got a problem, and it's that I can't see the difference of one of the definitions of Noetherian ring, and supposing the zorn lemma for that ring. I mean, if I use the definition of the maximal ...
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Closure of a set in a “Topology of finite complement”

Well, I was reading this article by Kelley and when reached the point where he say that $X_a$ is closed in $Y_a$ I had to stop, probably mine is just a stupid misunderstand but can't figure out how to ...
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Did I use axiom of choice in my proof?

I have two different affine open covers for a scheme $X$, say $X = \cup_{i \in I} U_i$ and $X = \cup_{j \in J} V_j$. For each $p \in X$, we know there exist some $i(p)$ and $j(p)$ such that $p \in ...
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Every epimorphism in Sets is split: why is it equivalent to axiom of choice?

Suppose that $f: A \rightarrow B$ is epic in Sets. One can construct a section $s: B \rightarrow A$ of $f$ as follow: Let us define an equivalence relation $R$ on $A$ as follow: $aRa'$ iff $a, a' \in ...
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Forcing reference

Who first proved that, over ZF, the statement (1) The reals are well-orderable is strictly stronger than the statement (2) Every real-indexed family of nonempty sets of reals has a choice ...